# Properties

 Label 418.2.a.c Level $418$ Weight $2$ Character orbit 418.a Self dual yes Analytic conductor $3.338$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 + 3 * q^3 + q^4 - 2 * q^5 + 3 * q^6 + q^7 + q^8 + 6 * q^9 $$q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9} - 2 q^{10} + q^{11} + 3 q^{12} - 7 q^{13} + q^{14} - 6 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} + q^{19} - 2 q^{20} + 3 q^{21} + q^{22} + 3 q^{23} + 3 q^{24} - q^{25} - 7 q^{26} + 9 q^{27} + q^{28} + q^{29} - 6 q^{30} + 2 q^{31} + q^{32} + 3 q^{33} - 3 q^{34} - 2 q^{35} + 6 q^{36} - 6 q^{37} + q^{38} - 21 q^{39} - 2 q^{40} - 2 q^{41} + 3 q^{42} + 4 q^{43} + q^{44} - 12 q^{45} + 3 q^{46} + 3 q^{48} - 6 q^{49} - q^{50} - 9 q^{51} - 7 q^{52} + 3 q^{53} + 9 q^{54} - 2 q^{55} + q^{56} + 3 q^{57} + q^{58} + 7 q^{59} - 6 q^{60} - 12 q^{61} + 2 q^{62} + 6 q^{63} + q^{64} + 14 q^{65} + 3 q^{66} + 15 q^{67} - 3 q^{68} + 9 q^{69} - 2 q^{70} + 6 q^{71} + 6 q^{72} - 9 q^{73} - 6 q^{74} - 3 q^{75} + q^{76} + q^{77} - 21 q^{78} - 8 q^{79} - 2 q^{80} + 9 q^{81} - 2 q^{82} + 16 q^{83} + 3 q^{84} + 6 q^{85} + 4 q^{86} + 3 q^{87} + q^{88} - 16 q^{89} - 12 q^{90} - 7 q^{91} + 3 q^{92} + 6 q^{93} - 2 q^{95} + 3 q^{96} + 8 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100})$$ q + q^2 + 3 * q^3 + q^4 - 2 * q^5 + 3 * q^6 + q^7 + q^8 + 6 * q^9 - 2 * q^10 + q^11 + 3 * q^12 - 7 * q^13 + q^14 - 6 * q^15 + q^16 - 3 * q^17 + 6 * q^18 + q^19 - 2 * q^20 + 3 * q^21 + q^22 + 3 * q^23 + 3 * q^24 - q^25 - 7 * q^26 + 9 * q^27 + q^28 + q^29 - 6 * q^30 + 2 * q^31 + q^32 + 3 * q^33 - 3 * q^34 - 2 * q^35 + 6 * q^36 - 6 * q^37 + q^38 - 21 * q^39 - 2 * q^40 - 2 * q^41 + 3 * q^42 + 4 * q^43 + q^44 - 12 * q^45 + 3 * q^46 + 3 * q^48 - 6 * q^49 - q^50 - 9 * q^51 - 7 * q^52 + 3 * q^53 + 9 * q^54 - 2 * q^55 + q^56 + 3 * q^57 + q^58 + 7 * q^59 - 6 * q^60 - 12 * q^61 + 2 * q^62 + 6 * q^63 + q^64 + 14 * q^65 + 3 * q^66 + 15 * q^67 - 3 * q^68 + 9 * q^69 - 2 * q^70 + 6 * q^71 + 6 * q^72 - 9 * q^73 - 6 * q^74 - 3 * q^75 + q^76 + q^77 - 21 * q^78 - 8 * q^79 - 2 * q^80 + 9 * q^81 - 2 * q^82 + 16 * q^83 + 3 * q^84 + 6 * q^85 + 4 * q^86 + 3 * q^87 + q^88 - 16 * q^89 - 12 * q^90 - 7 * q^91 + 3 * q^92 + 6 * q^93 - 2 * q^95 + 3 * q^96 + 8 * q^97 - 6 * q^98 + 6 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 3.00000 1.00000 −2.00000 3.00000 1.00000 1.00000 6.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.a.c 1
3.b odd 2 1 3762.2.a.j 1
4.b odd 2 1 3344.2.a.a 1
11.b odd 2 1 4598.2.a.j 1
19.b odd 2 1 7942.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.c 1 1.a even 1 1 trivial
3344.2.a.a 1 4.b odd 2 1
3762.2.a.j 1 3.b odd 2 1
4598.2.a.j 1 11.b odd 2 1
7942.2.a.a 1 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(418))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 3$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T + 7$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T - 3$$
$29$ $$T - 1$$
$31$ $$T - 2$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T - 7$$
$61$ $$T + 12$$
$67$ $$T - 15$$
$71$ $$T - 6$$
$73$ $$T + 9$$
$79$ $$T + 8$$
$83$ $$T - 16$$
$89$ $$T + 16$$
$97$ $$T - 8$$